src/numerical/slatec/fortran/dqags.f

   1 *DECK DQAGS
   2       SUBROUTINE DQAGS (F, A, B, EPSABS, EPSREL, RESULT, ABSERR, NEVAL,
   3      +   IER, LIMIT, LENW, LAST, IWORK, WORK)
   4 C***BEGIN PROLOGUE  DQAGS
   5 C***PURPOSE  The routine calculates an approximation result to a given
   6 C            Definite integral  I = Integral of F over (A,B),
   7 C            Hopefully satisfying following claim for accuracy
   8 C            ABS(I-RESULT).LE.MAX(EPSABS,EPSREL*ABS(I)).
   9 C***LIBRARY   SLATEC (QUADPACK)
  10 C***CATEGORY  H2A1A1
  11 C***TYPE      DOUBLE PRECISION (QAGS-S, DQAGS-D)
  12 C***KEYWORDS  AUTOMATIC INTEGRATOR, END POINT SINGULARITIES,
  13 C             EXTRAPOLATION, GENERAL-PURPOSE, GLOBALLY ADAPTIVE,
  14 C             QUADPACK, QUADRATURE
  15 C***AUTHOR  Piessens, Robert
  16 C             Applied Mathematics and Programming Division
  17 C             K. U. Leuven
  18 C           de Doncker, Elise
  19 C             Applied Mathematics and Programming Division
  20 C             K. U. Leuven
  21 C***DESCRIPTION
  22 C
  23 C        Computation of a definite integral
  24 C        Standard fortran subroutine
  25 C        Double precision version
  26 C
  27 C
  28 C        PARAMETERS
  29 C         ON ENTRY
  30 C            F      - Double precision
  31 C                     Function subprogram defining the integrand
  32 C                     Function F(X). The actual name for F needs to be
  33 C                     Declared E X T E R N A L in the driver program.
  34 C
  35 C            A      - Double precision
  36 C                     Lower limit of integration
  37 C
  38 C            B      - Double precision
  39 C                     Upper limit of integration
  40 C
  41 C            EPSABS - Double precision
  42 C                     Absolute accuracy requested
  43 C            EPSREL - Double precision
  44 C                     Relative accuracy requested
  45 C                     If  EPSABS.LE.0
  46 C                     And EPSREL.LT.MAX(50*REL.MACH.ACC.,0.5D-28),
  47 C                     The routine will end with IER = 6.
  48 C
  49 C         ON RETURN
  50 C            RESULT - Double precision
  51 C                     Approximation to the integral
  52 C
  53 C            ABSERR - Double precision
  54 C                     Estimate of the modulus of the absolute error,
  55 C                     which should equal or exceed ABS(I-RESULT)
  56 C
  57 C            NEVAL  - Integer
  58 C                     Number of integrand evaluations
  59 C
  60 C            IER    - Integer
  61 C                     IER = 0 Normal and reliable termination of the
  62 C                             routine. It is assumed that the requested
  63 C                             accuracy has been achieved.
  64 C                     IER.GT.0 Abnormal termination of the routine
  65 C                             The estimates for integral and error are
  66 C                             less reliable. It is assumed that the
  67 C                             requested accuracy has not been achieved.
  68 C            ERROR MESSAGES
  69 C                     IER = 1 Maximum number of subdivisions allowed
  70 C                             has been achieved. One can allow more sub-
  71 C                             divisions by increasing the value of LIMIT
  72 C                             (and taking the according dimension
  73 C                             adjustments into account. However, if
  74 C                             this yields no improvement it is advised
  75 C                             to analyze the integrand in order to
  76 C                             determine the integration difficulties. If
  77 C                             the position of a local difficulty can be
  78 C                             determined (E.G. SINGULARITY,
  79 C                             DISCONTINUITY WITHIN THE INTERVAL) one
  80 C                             will probably gain from splitting up the
  81 C                             interval at this point and calling the
  82 C                             integrator on the subranges. If possible,
  83 C                             an appropriate special-purpose integrator
  84 C                             should be used, which is designed for
  85 C                             handling the type of difficulty involved.
  86 C                         = 2 The occurrence of roundoff error is detec-
  87 C                             ted, which prevents the requested
  88 C                             tolerance from being achieved.
  89 C                             The error may be under-estimated.
  90 C                         = 3 Extremely bad integrand behaviour
  91 C                             occurs at some points of the integration
  92 C                             interval.
  93 C                         = 4 The algorithm does not converge.
  94 C                             Roundoff error is detected in the
  95 C                             Extrapolation table. It is presumed that
  96 C                             the requested tolerance cannot be
  97 C                             achieved, and that the returned result is
  98 C                             the best which can be obtained.
  99 C                         = 5 The integral is probably divergent, or
 100 C                             slowly convergent. It must be noted that
 101 C                             divergence can occur with any other value
 102 C                             of IER.
 103 C                         = 6 The input is invalid, because
 104 C                             (EPSABS.LE.0 AND
 105 C                              EPSREL.LT.MAX(50*REL.MACH.ACC.,0.5D-28)
 106 C                             OR LIMIT.LT.1 OR LENW.LT.LIMIT*4.
 107 C                             RESULT, ABSERR, NEVAL, LAST are set to
 108 C                             zero.  Except when LIMIT or LENW is
 109 C                             invalid, IWORK(1), WORK(LIMIT*2+1) and
 110 C                             WORK(LIMIT*3+1) are set to zero, WORK(1)
 111 C                             is set to A and WORK(LIMIT+1) TO B.
 112 C
 113 C         DIMENSIONING PARAMETERS
 114 C            LIMIT - Integer
 115 C                    DIMENSIONING PARAMETER FOR IWORK
 116 C                    LIMIT determines the maximum number of subintervals
 117 C                    in the partition of the given integration interval
 118 C                    (A,B), LIMIT.GE.1.
 119 C                    IF LIMIT.LT.1, the routine will end with IER = 6.
 120 C
 121 C            LENW  - Integer
 122 C                    DIMENSIONING PARAMETER FOR WORK
 123 C                    LENW must be at least LIMIT*4.
 124 C                    If LENW.LT.LIMIT*4, the routine will end
 125 C                    with IER = 6.
 126 C
 127 C            LAST  - Integer
 128 C                    On return, LAST equals the number of subintervals
 129 C                    produced in the subdivision process, determines the
 130 C                    number of significant elements actually in the WORK
 131 C                    Arrays.
 132 C
 133 C         WORK ARRAYS
 134 C            IWORK - Integer
 135 C                    Vector of dimension at least LIMIT, the first K
 136 C                    elements of which contain pointers
 137 C                    to the error estimates over the subintervals
 138 C                    such that WORK(LIMIT*3+IWORK(1)),... ,
 139 C                    WORK(LIMIT*3+IWORK(K)) form a decreasing
 140 C                    sequence, with K = LAST IF LAST.LE.(LIMIT/2+2),
 141 C                    and K = LIMIT+1-LAST otherwise
 142 C
 143 C            WORK  - Double precision
 144 C                    Vector of dimension at least LENW
 145 C                    on return
 146 C                    WORK(1), ..., WORK(LAST) contain the left
 147 C                     end-points of the subintervals in the
 148 C                     partition of (A,B),
 149 C                    WORK(LIMIT+1), ..., WORK(LIMIT+LAST) contain
 150 C                     the right end-points,
 151 C                    WORK(LIMIT*2+1), ..., WORK(LIMIT*2+LAST) contain
 152 C                     the integral approximations over the subintervals,
 153 C                    WORK(LIMIT*3+1), ..., WORK(LIMIT*3+LAST)
 154 C                     contain the error estimates.
 155 C
 156 C***REFERENCES  (NONE)
 157 C***ROUTINES CALLED  DQAGSE, XERMSG
 158 C***REVISION HISTORY  (YYMMDD)
 159 C   800101  DATE WRITTEN
 160 C   890831  Modified array declarations.  (WRB)
 161 C   890831  REVISION DATE from Version 3.2
 162 C   891214  Prologue converted to Version 4.0 format.  (BAB)
 163 C   900315  CALLs to XERROR changed to CALLs to XERMSG.  (THJ)
 164 C***END PROLOGUE  DQAGS
 165 C
 166 C
 167       DOUBLE PRECISION A,ABSERR,B,EPSABS,EPSREL,F,RESULT,WORK
 168       INTEGER IER,IWORK,LAST,LENW,LIMIT,LVL,L1,L2,L3,NEVAL
 169 C
 170       DIMENSION IWORK(*),WORK(*)
 171 C
 172       EXTERNAL F
 173 C
 174 C         CHECK VALIDITY OF LIMIT AND LENW.
 175 C
 176 C***FIRST EXECUTABLE STATEMENT  DQAGS
 177       IER = 6
 178       NEVAL = 0
 179       LAST = 0
 180       RESULT = 0.0D+00
 181       ABSERR = 0.0D+00
 182       IF(LIMIT.LT.1.OR.LENW.LT.LIMIT*4) GO TO 10
 183 C
 184 C         PREPARE CALL FOR DQAGSE.
 185 C
 186       L1 = LIMIT+1
 187       L2 = LIMIT+L1
 188       L3 = LIMIT+L2
 189 C
 190       CALL DQAGSE(F,A,B,EPSABS,EPSREL,LIMIT,RESULT,ABSERR,NEVAL,
 191      1  IER,WORK(1),WORK(L1),WORK(L2),WORK(L3),IWORK,LAST)
 192 C
 193 C         CALL ERROR HANDLER IF NECESSARY.
 194 C
 195       LVL = 0
 196 10    IF(IER.EQ.6) LVL = 1
 197       IF (IER .NE. 0) CALL XERMSG ('SLATEC', 'DQAGS',
 198      +   'ABNORMAL RETURN', IER, LVL)
 199       RETURN
 200       END